This already exists. For sending money to a friends account you just log in to the bank website, type their bank account number and the amount and voila.
For sending money to companies there is something called iDEAL. On the company's website you click checkout and you get sent to your bank's website where you get a page with the amount and a button to confirm. Unfortunately this iDEAL system is currently only available in the Netherlands.
account # is way too long, I meant something that is tied to an email address
Money bookers use the email address as identifier (at least some years ago when I used it). http://www.moneybookers.com
Assuming you've given that friend money before you can just type his name.
I'm pretty sure you can in Canada, using the interac (debit) network:
http://www.interac.ca/consumers/productsandservices_ol_emt.p...
you can use your ATM card at any bank...so banks can work together...they should just create bankpal and have it be a service you can use to send money to your friend's account